Band 70, Heft 5

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.

This paper is devoted to the study of the class of decomposable double MS -algebras. Necessary and sufficient conditions for a decomposable MS -algebra to be a decomposable double MS -algebra are deduced. We construct decomposable double MS -algebras by means of decomposable MS -quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS -algebras and decomposable MS -quadruples. Moreover, a construction of decomposable K 2 -algebras (Stone algebras) by means of K 2 -quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS -algebras in terms of decomposable MS -quadruples.

In this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence.

In this paper, we find all the solutions ( X , Y , Z ) = ( F I , F J , F K ), where F I , F J , and F K represent nonzero Fibonacci numbers, satisfying a generalization of Markoff equation called the Jin-Schmidt equation: AX 2 + BY 2 + CZ 2 = DXYZ + 1.

Suppose f is a degree d polynomial with integer coefficients whose derivative f ′ is a polynomial reducible over ℚ. We give a lower bound for the distance between two distinct roots of f in terms of d , the height H ( f ) of f , and the degree m of the irreducible factor of f ′ with largest degree. The exponent ( d + m − 1)/2 that appears as the power of H ( f ) is smaller than the corresponding exponent d − 1 in Mahler’s bound.

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely: (va+(1− v)b)2− v∑ j=1N2j(b− ab2j− 12j)2≤ (avb1− v)2+v2(a− b)2$$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j}]{ab^{2^{j}-1} }\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}(a-b)^{2} \end{array}$$ for v ∉ [0, 12N+1$\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$], N ∈ ℕ, a , b > 0; and (va+(1− v)b)2− (1− v)∑ j=1N2j(a− a2j− 1b2j)2≤ (avb1− v)2+(1− v)2(a− b)2$$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+(1-v)^{2}(a-b)^{2} \end{array}$$ for v ∉ [1 − 12N+1$\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$, 1], N ∈ ℕ, a , b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

In the article, we present the best possible parameters α 1 , β 1 , α 2 , β 2 ∈ ℝ and α 3 , β 3 ∈ [1/2, 1] such that the double inequalities α 1C(a,b)+(1− α 1)A(a,b)< T3(a,b)< β 1C(a,b)+(1− β 1)A(a,b),α 2C(a,b)+(1− α 2)Q(a,b)< T3(a,b)< β 2C(a,b)+(1− β 2)Q(a,b),C(α 3;a,b)< T3(a,b)< C(β 3;a,b)$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$ hold for a , b > 0 with a ≠ b , and provide new bounds for the complete elliptic integral of the second kind, where A ( a , b ) = ( a + b )/2 is the arithmetic mean, Q(a,b)=a2+b2/2$\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C ( a , b ) = ( a 2 + b 2 )/( a + b ) is the contra-harmonic mean, C ( p ; a , b ) = C [ pa + (1 – p ) b , pb + (1 – p ) a ] is the one-parameter contra-harmonic mean and T3(a,b)=(2π ∫ 0π /2a3cos2⁡ θ +b3sin2⁡ θ dθ )2/3$\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an integration over sets yields a new deficient topological measure if we integrate a nonnegative continuous vanishing at infinity function; and it produces a signed deficient topological measure if we integrate a continuous function on a compact space. We present many properties of these resulting deficient topological measures and of signed deficient topological measures. In particular, they are absolutely continuous with respect to the original deficient topological measure, and their corresponding non-linear functionals are Lipschitz continuous. Deficient topological measures obtained by integration over sets can also be obtained from non-linear functionals. We show that for a deficient topological measure μ that assumes finitely many values, there is a function f such that ∫ X$\begin{array}{} \int\limits_X \end{array}$ f d μ = 0, but ∫ X$\begin{array}{} \int\limits_X \end{array}$ (– f ) d μ ≠ 0. We present different criteria for ∫ X$\begin{array}{} \int\limits_X \end{array}$ f d μ = 0. We also prove some convergence results, including a Monotone convergence theorem.

The aim of this paper is to generalize several differential subordination results, involving arithmetic, geometric and harmonic means of the expressions p ( z ) and p(z)+zp′(z)p(z).$\begin{array}{} p(z)+\frac{zp'(z)}{p(z)}. \end{array} $ Are also given certain applications of the main results.

In this paper, we present necessary and sufficient conditions of boundedness of L -index in joint variables for vector-valued functions analytic in the unit ball B2={z∈ C2:|z|=|z1|2+|z2|2< 1},$\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = ( l 1 , l 2 ): 𝔹 2 → R+2$\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function. Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L -index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of ( p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L ( z ).

In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form (r(t)(y(t)+p(t)y(t− τ ))″)″+q(t)G(y(t− α ))− h(t)H(y(t− σ ))=0$$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0∞ tr(t)d  t=∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$

The aim in this work is to investigate oscillation criteria for a class of nonlinear discrete fractional order equations with damping term of the form Δ a(t)Δ r(t)gΔ α x(t)β +p(t)Δ r(t)gΔ α x(t)β +F(t,G(t))=0,t∈ Nt0.$$\begin{array}{} \displaystyle \Delta\left[a(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta\right]+p(t)\left[\Delta\left(r(t)g\left(\Delta^\alpha x(t)\right)\right)\right]^\beta+F(t,G(t))=0, t\in N_{t_0}. \end{array}$$ In the above equation α (0 < α ≤ 1) is the fractional order, G(t)=∑ s=t0t− 1+α t− s− 1(− α )x(s)$\begin{array}{} \displaystyle G(t)=\sum\limits_{s=t_0}^{t-1+\alpha}\left(t-s-1\right)^{(-\alpha)}x(s) \end{array}$ and Δ α is the difference operator of the Riemann-Liouville (R-L) derivative of order α . We establish some new sufficient conditions for the oscillation of fractional order difference equations with damping term based on a Riccati transformation technique and some inequalities. We provide numerical examples to illustrate the validity of the theoretical results.

In [Rahimi, M.: Entropy as an integral operator , Math. Slovaca 69 (1) (2019), 139–146], we assigned an integral operator on a Hilbert space to any topological dynamical system of finite entropy and stated the entropy of the system in terms of the spectrum of the defined operator. Unfortunately, there is a mistake in the proof of the main theorem of the paper which makes the result incorrect. So, we can not extract the entropy of a topological dynamical system in terms of the spectrum of the introduced operator. In this note, we modify the main theorem of [11] by giving a modification to the proof of the theorem. Then, replacing the integral operator introduced in [11] by another linear operator, we will state the entropy of the system in terms of the spectrum of the new operator.

In this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l -groups. For instance we show that for an Archimedean Riesz space G with order unit u , the unital topology and the strong link topology are the same.

We prove that the topological complexity of a quaternionic flag manifold is half of its real dimension. For the real oriented Grassmann manifolds G͠ n , k , 3 ≤ k ≤ [ n /2], the zero-divisor cup-length of the rational cohomology of G͠ n , k is computed in terms of n and k which gives a lower bound for the topological complexity of G͠ n , k , TC ( G͠ n , k ). When k = 3, it is observed in certain cases that better lower bounds for TC ( G͠ n ,3 ) are obtained using ℤ 2 -cohomology.

In this paper, a bivariate extension of exponentiated Fréchet distribution is introduced, namely a bivariate exponentiated Fréchet (BvEF) distribution whose marginals are univariate exponentiated Fréchet distribution. Several properties of the proposed distribution are discussed, such as the joint survival function, joint probability density function, marginal probability density function, conditional probability density function, moments, marginal and bivariate moment generating functions. Moreover, the proposed distribution is obtained by the Marshall-Olkin survival copula. Estimation of the parameters is investigated by the maximum likelihood with the observed information matrix. In addition to the maximum likelihood estimation method, we consider the Bayesian inference and least square estimation and compare these three methodologies for the BvEF. A simulation study is carried out to compare the performance of the estimators by the presented estimation methods. The proposed bivariate distribution with other related bivariate distributions are fitted to a real-life paired data set. It is shown that, the BvEF distribution has a superior performance among the compared distributions using several tests of goodness–of–fit.

In this paper, we mainly consider the existence and Hyers-Ulam stability of solutions for a class of fractional differential equations involving time-varying delays and non-instantaneous impulses. By the Krasnoselskii’s fixed point theorem, we present the new constructive existence results for the addressed equation. In addition, we deduce that the equations have Hyers-Ulam stable solutions by utilizing generalized Grönwall’s inequality. Some results in this literature are new and improve some early conclusions.

We fix z 0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜 z 0 := the field of germs of meromorphic functions at z 0 . We fix f 1 , …, f r ∈ 𝓜 z 0 and we consider the 𝔽-algebras S := 𝔽[ f 1 , …, f r ] and S¯ :=F[f1± 1,… ,fr± 1].$\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings Shol:=F[fa:=f1a1⋯ frar:(a1,… ,ar)∈ Nr and fa is holomorphic at z0],S¯ hol:=F[fa:=f1a1⋯ frar:(a1,… ,ar)∈ Zr and fa is holomorphic at z0],$$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$ and we tackle in detail the case 𝔽 = 𝓜 <1 , the field of meromorphic functions of order < 1, and f j ’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.